No. Healthy babies naturally swallow or cough up fluids—it's a reflex all people have to make sure their airway is kept clear.
Babies might actually clear such fluids better when on their backs because of the location of the windpipe (trachea) when in the back sleep position.
When the baby is in the back sleep position, the windpipe lies on top of the esophagus, which leads to the stomach. Anything regurgitated or refluxed from the stomach through the esophagus has to work against gravity to enter the trachea and cause choking. When the baby is sleeping on its stomach, such fluids will exit the esophagus and pool at the opening for the trachea, making choking much more likely.
Cases of fatal choking are very rare except when related to a medical condition. The number of fatal choking deaths has not increased since back sleeping recommendations began. In most of the few reported cases of fatal choking, an infant was sleeping on his or her stomach.